# Addition and Multiplication Properties for Real Numbers

Contributed by:
This pdf includes the following topics:-
and Multiplication
Commutative Properties
Associative Properties
Using Properties to Simplify Expressions
Examples
1. English Spanish
1.3
and Multiplication
Does the order in which
you perform an operation matter?
1 ACTIVITY: Does Order Matter?
Work with a partner. Place each statement in the correct oval.
a. Fasten 5 shirt buttons. b. Put on a shirt and tie.
c. Fill and seal an envelope. d. Floss your teeth.
e. Put on your shoes. f. Chew and swallow.
Order Matters Order Doesn’t Matter
Think of some math problems using the four operations where order
matters and some where order doesn’t matter.
Commute
When you commute the positions you switch their positions.
of two stuffed animals on a shelf,
2 ACTIVITY: Commutative Properties
Work with a partner.
a. Which of the following are true?
? ?
3+5=5+3 3−5=5−3
? ?
9×3=3×9 9÷3=3÷9
b. The true equations show the Commutative Properties of Addition and
Multiplication. Why are they called “commutative?” Write the properties.
14 Chapter 1 Expressions and Number Properties
2. English Spanish
Associate
You have two best friends. Sometimes And sometimes you associate
you associate with one of them. with the other.
3 ACTIVITY: Associative Properties
Work with a partner.
a. Which of the following are true?
? ?
8 + (3 + 1) = (8 + 3) + 1 8 − (3 − 1) = (8 − 3) − 1
? ?
12 × (6 × 2) = (12 × 6) × 2 12 ÷ (6 ÷ 2) = (12 ÷ 6) ÷ 2
b. The true equations show the Associative Properties of Addition and
Multiplication. Why are they called “associative?” Write the properties.
4. IN YOUR OWN WORDS Does the order in which you perform an
operation matter?
5. MENTAL MATH Explain how you can use the Commutative and
11 + 7 + 12 + 13 + 8 + 9
6. SECRET CODE The creatures on a distant planet use the symbols
■ , ◆ , ★ , and ● for the four operations.
a. Use the codes to decide which symbol represents addition and which
symbol represents multiplication. Explain your reasoning.
3● 4=4● 3
3★4=4★3
2 ● (5 ● 3) = (2 ● 5) ● 3
2 ★ (5 ★ 3) = (2 ★ 5) ★ 3
0● 4=0
0★4=4
b. Make up your own symbols for addition and multiplication. Write codes
represents addition and which symbol represents multiplication.
multiplication to complete Exercises 5 – 8 on page 18.
Section 1.3 Properties of Addition and Multiplication 15
3. English Spanish
1.3 Lesson
Lesson Tutorials
Expressions with the same value, like 12 + 7 and 7 + 12, are equivalent
Key Vocabulary
expressions. The commutative and associative properties can be used to
equivalent write equivalent expressions.
expressions, p. 16
Commutative Properties
Words Changing the order of addends or factors does not change the
sum or product.
Numbers 5+8=8+5 Algebra a+b=b+a

5 8=8 5 ⋅ ⋅
a b=b a ⋅
Associative Properties
Words Changing the grouping of addends or factors does not change
the sum or product.
Numbers (7 + 4) + 2 = 7 + (4 + 2)
⋅ ⋅
(7 4) 2 = 7 (4 2)⋅ ⋅
Algebra (a + b) + c = a + (b + c)
⋅ ⋅
(a b) c = a (b c)⋅ ⋅
EXAMPLE 1 Using Properties to Simplify Expressions
a. Simplify the expression 7 + (12 + x).
7 + (12 + x) = (7 + 12) + x Associative Property of Addition
= 19 + x Add 7 and 12.
b. Simplify the expression (6.1 + x) + 8.4.
(6.1 + x) + 8.4 = (x + 6.1) + 8.4 Commutative Property of Addition
= x + (6.1 + 8.4) Associative Property of Addition
= x + 14.5 Add 6.1 and 8.4.
c. Simplify the expression 5(11y).

5(11y) = (5 11)y Associative Property of Multiplication
= 55y Multiply 5 and 11.
Simplify the expression. Explain each step.
Exercises 5 – 8 1. 10 + (a + 9) 2. (c + 25.3) + 17.9 3. 5(4n)
16 Chapter 1 Expressions and Number Properties
4. English Spanish
Words The sum of any number and 0 is that number.
Numbers 7+0=7 Algebra a+0=a
Multiplication Properties of Zero and One
Words The product of any number and 0 is 0.
The product of any number and 1 is that number.
Numbers 9×0=0 Algebra ⋅
a 0=0
4×1=4 a ⋅1 = a
EXAMPLE 2 Using Properties to Simplify Expressions
a. Simplify the expression 9 0 p.⋅ ⋅
⋅ ⋅ ⋅ ⋅
9 0 p = (9 0) p Associative Property of Multiplication
= 0 ⋅p = 0 Multiplication Property of Zero
⋅⋅
b. Simplify the expression 4.5 r 1.
⋅⋅ ⋅ ⋅
4.5 r 1 = 4.5 (r 1) Associative Property of Multiplication
= 4.5 ⋅ r Multiplication Property of One
= 4.5r
EXAMPLE 3 Real-Life Application
You and six friends play on a basketball team. A sponsor paid \$100
for the league fee, x dollars for each player’s T-shirt, and \$68.25 for
Common Error trophies. Write an expression for the total amount paid by the sponsor.
You and six friends are
Add the entry fee, the cost of the T-shirts, and the cost of the trophies.
on the team, so use the
expression 7x, not 6x, 100 + 7x + 68.25 = 7x + 100 + 68.25 Commutative Property of Addition
to represent the cost of
the T-shirts. = 7x + 168.25 Add 100 and 68.25.
An expression for the total amount is 7x + 168.25.
Simplify the expression. Explain each step.
Exercises 9 – 23 ⋅ ⋅
4. 12 b 0 5. ⋅ ⋅
1 m 24 6. (t + 15) + 0
7. WHAT IF? In Example 3, your sponsor paid \$54.75 for trophies.
Write an expression for the total amount paid by the sponsor.
Section 1.3 Properties of Addition and Multiplication 17
5. English Spanish
1.3 Exercises
Help with Homework
1. NUMBER SENSE Write an example of a sum of fractions. Show that the
Commutative Property of Addition is true for the sum.
2. OPEN-ENDED Write an algebraic expression that can be simplified using the
3. OPEN-ENDED Write an algebraic expression that can be simplified using the
Associative Property of Multiplication and the Multiplication Property of One.
4. WHICH ONE DOESN’T BELONG? Which statement does not belong with the
7 + (x + 4) = 7 + (4 + x) (3 + b) + 2 = (b + 3) + 2
9 + (7 + w) = (9 + 7) + w (4 + n) + 6 = (n + 4) + 6
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Tell which property is illustrated by the statement.
1 ⋅ ⋅
5. 5 p = p 5 6. 2 + (12 + r) = (2 + 12) + r
7. 4 ⋅ (x ⋅ 10) = (4 ⋅ x) ⋅ 10 8. x + 7.5 = 7.5 + x
2 9. (c + 2) + 0 = c + 2 ⋅
10. a 1 = a

11. ERROR ANALYSIS Describe and correct the
error in stating the property illustrated (7 + x) + 3 = (x + 7 ) + 3
by the statement. Associative Property of Addition
Simplify the expression. Explain each step.
12. 6 + (5 + x) 13. (14 + y) + 3 14. 6(2b)
15. 7(9w) 16. 3.2 + (x + 5.1) 17. (0 + a) + 8
18. 9 c 4 ⋅⋅ ⋅ ⋅
19. (18.6 d) 1 20. (3k + 4.2) + 8.6
21. (2.4 + 4n) + 9 22. (3s) ⋅ 8 ⋅ ⋅
23. z 0 12
24. GEOMETRY The expression 12 + x + 4 represents
the perimeter of a triangle. Simplify the expression.
25. SCOUT COOKIES A case of Scout cookies has 10 cartons. A carton has
12 boxes. The amount you make on a whole case is 10(12x) dollars.
a. What does x represent?
b. Simplify the expression.
18 Chapter 1 Expressions and Number Properties
6. English Spanish
26. GEOMETRY The volume of the rectangular prism is 12.5 x 1. ⋅⋅ 1
a. Simplify the expression.
x
b. Match x = 0.25, 12.5, and 144 with the object. 12.5
A. siding for a house B. ruler C. square floor tile
Write the phrase as an expression. Then, simplify the expression.
27. 7 plus the sum of a number x and 5
28. the product of 8 and a number y multiplied by 9
Copy and complete the statement using the specified property.
Property Statement
29. Associative Property of Multiplication 7(2y) =
30. Commutative Property of Multiplication ⋅ ⋅
13.2 (x 1) =
31. Associative Property of Addition 17 + (6 + 2x) =
32. Addition Property of Zero 2 + (c + 0) =
33. Multiplication Property of One ⋅ ⋅
1 w 16 =
34. HATS You and a friend sell hats at a fair booth. You sell 16 hats on the first
shift and 21 hats on the third shift. Your friend sells x hats on the second shift.
a. Write an expression for the number of hats sold.
b. The expression 37(14) + 10x represents the amount made for both
of you. How can you tell that your friend was selling the hats for a
discounted price?
c. You took in more money than your friend. What can you say
Evaluate the expression. SKILLS REVIEW HANDBOOK
35. 7(10 + 4) 36. 12(10 − 1) 37. 6(5 + 10) 38. 8(30 − 5)
Find the prime factorization of the number. SKILLS REVIEW HANDBOOK
39. 37 40. 144 41. 147 42. 205
43. MULTIPLE CHOICE A bag has 16 blue, 20 red, and 24 green marbles. What
fraction of the marbles in the bag are blue? SKILLS REVIEW HANDBOOK
1 4 4 11

A — ○
B — ○
C — ○
D —
5 15 11 15
Section 1.3 Properties of Addition and Multiplication 19